Bose-Einstein condensation in dilute gases

585 Pages·2008·3.919 MB·English

by  C. J. Pethick, H. Smith

BOSE–EINSTEIN CONDENSATION
IN DILUTE GASES
Second Edition
Since an atomic Bose–Einstein condensate, predicted by Einstein in 1925, was
first produced in the laboratory in 1995, the study of ultracold Bose and Fermi
gases has become one of the most active areas in contemporary physics. In
this book the authors explain phenomena in ultracold gases from basic prin-
ciples, without assuming a detailed knowledge of atomic and condensed mat-
ter physics. This new edition has been revised and updated, and includes new
chapters on optical lattices, low dimensions, and strongly interacting Fermi
systems.
This book provides a unified introduction to the physics of ultracold atomic
Bose and Fermi gases at advanced undergraduate and graduate levels. The book
will also be of interest to anyone with a general background in physics, from
undergraduates to researchers in the field. Chapters cover the statistical phys-
ics of trapped gases, atomic properties, cooling and trapping atoms, interatomic
interactions,structureoftrappedcondensates,collectivemodes,rotatingcondens-
ates,superfluidity,mixturesandspinorcondensates,interferenceandcorrelations,
optical lattice low-dimensional systems, Fermi gases and the crossover from the
BCS superfluid state to a Bose–Einstein condensate of diatomic molecules. Prob-
lems are included at the end of each chapter.
Christopher Pethick graduated with a D.Phil. in 1965 from the University
of Oxford, and he had a research fellowship there until 1970. During the years
1966–69 he was a postdoctoral fellow at the University of Illinois at Urbana–
Champaign, where he joined the faculty in 1970, becoming Professor of Physics
in 1973. Following periods spent at the Landau Institute for Theoretical Physics,
Moscow and at Nordita (Nordic Institute for Theoretical Physics), Copenhagen,
as a visiting scientist, he accepted a permanent position at Nordita in 1975, and
divided his time for many years between Nordita and the University of Illinois.
Apart from the subject of the present book, Professor Pethick’s main research
interests are condensed matter physics (quantum liquids, especially 3He, 4He and
superconductors)andastrophysics(particularlythepropertiesofdensematterand
the interiors of neutron stars). He is also the co-author of Landau Fermi-Liquid
Theory: Concepts and Applications (1991).
Henrik Smith obtained his mag. scient. degree in 1966 from the University of
Copenhagen and spent the next few years as a postdoctoral fellow at Cornell
University and as a visiting scientist at the Institute for Theoretical Physics,
Helsinki. In 1972 he joined the faculty of the University of Copenhagen where
he became dr. phil. in 1977 and Professor of Physics in 1978. He has also
workedasaguestscientistattheBellLaboratories,NewJersey.ProfessorSmith’s
researchfieldiscondensedmatterphysicsandlow-temperaturephysics,including
quantum liquids and the properties of superfluid 3He transport properties of
normal and superconducting metals and two-dimensional electron systems. His
other books include Transport Phenomena (1989) and Introduction to Quantum
Mechanics (1991).
Thetwoauthorshaveworkedtogetheronproblemsinlow-temperaturephysics,
in particular on the superfluid phases of liquid 3He superconductors and dilute
quantum gases. This book derives from graduate-level lectures given by the
authors at the University of Copenhagen.
BOSE–EINSTEIN CONDENSATION
IN DILUTE GASES
Second Edition
C. J. PETHICK
Nordita
H. SMITH
UniversityofCopenhagen
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521846516
©C.PethickandH.Smith2008
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2001
SecondEdition2008
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-84651-6hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface page xiii
1 Introduction 1
1.1 Bose–Einstein condensation in atomic clouds 4
1.2 Superfluid 4He 7
1.3 Other condensates 9
1.4 Overview 10
Problems 15
References 15
2 The non-interacting Bose gas 17
2.1 The Bose distribution 17
2.1.1 Density of states 19
2.2 Transition temperature and condensate fraction 21
2.2.1 Condensate fraction 24
2.3 Density profile and velocity distribution 25
2.3.1 The semi-classical distribution 28
2.4 Thermodynamic quantities 33
2.4.1 Condensed phase 33
2.4.2 Normal phase 35
2.4.3 Specific heat close to T 36
c
2.5 Effect of finite particle number 38
Problems 39
References 40
3 Atomic properties 41
3.1 Atomic structure 41
3.2 The Zeeman effect 45
3.3 Response to an electric field 50
v
vi Contents
3.4 Energy scales 56
Problems 58
References 59
4 Trapping and cooling of atoms 60
4.1 Magnetic traps 61
4.1.1 The quadrupole trap 62
4.1.2 The TOP trap 64
4.1.3 Magnetic bottles and the Ioffe–Pritchard trap 66
4.1.4 Microtraps 69
4.2 Influence of laser light on an atom 71
4.2.1 Forces on an atom in a laser field 75
4.2.2 Optical traps 77
4.3 Laser cooling: the Doppler process 78
4.4 The magneto-optical trap 82
4.5 Sisyphus cooling 84
4.6 Evaporative cooling 96
4.7 Spin-polarized hydrogen 103
Problems 106
References 107
5 Interactions between atoms 109
5.1 Interatomic potentials and the van der Waals interaction 110
5.2 Basic scattering theory 114
5.2.1 Effective interactions and the scattering length 119
5.3 Scattering length for a model potential 125
5.4 Scattering between different internal states 130
5.4.1 Inelastic processes 135
5.4.2 Elastic scattering and Feshbach resonances 143
5.5 Determination of scattering lengths 151
5.5.1 Scattering lengths for alkali atoms and hydrogen 154
Problems 156
References 156
6 Theory of the condensed state 159
6.1 The Gross–Pitaevskii equation 159
6.2 The ground state for trapped bosons 162
6.2.1 A variational calculation 165
6.2.2 The Thomas–Fermi approximation 168
6.3 Surface structure of clouds 171
6.4 Healing of the condensate wave function 175
Contents vii
6.5 Condensates with dipolar interactions 176
Problems 179
References 180
7 Dynamics of the condensate 182
7.1 General formulation 182
7.1.1 The hydrodynamic equations 184
7.2 Elementary excitations 188
7.3 Collective modes in traps 196
7.3.1 Traps with spherical symmetry 197
7.3.2 Anisotropic traps 200
7.3.3 Collective coordinates and the variational method 204
7.4 Surface modes 211
7.5 Free expansion of the condensate 213
7.6 Solitons 215
7.6.1 Dark solitons 216
7.6.2 Bright solitons 222
Problems 223
References 224
8 Microscopic theory of the Bose gas 225
8.1 The uniform Bose gas 226
8.1.1 The Bogoliubov transformation 229
8.1.2 Elementary excitations 230
8.1.3 Depletion of the condensate 231
8.1.4 Ground-state energy 233
8.1.5 States with definite particle number 234
8.2 Excitations in a trapped gas 236
8.3 Non-zero temperature 241
8.3.1 The Hartree–Fock approximation 242
8.3.2 The Popov approximation 248
8.3.3 Excitations in non-uniform gases 250
8.3.4 The semi-classical approximation 251
References 253
9 Rotating condensates 255
9.1 Potential flow and quantized circulation 255
9.2 Structure of a single vortex 257
9.2.1 A vortex in a uniform medium 257
9.2.2 Vortices with multiple quanta of circulation 261
9.2.3 A vortex in a trapped cloud 262
viii Contents
9.2.4 An off-axis vortex 265
9.3 Equilibrium of rotating condensates 265
9.3.1 Traps with an axis of symmetry 266
9.3.2 Rotating traps 267
9.3.3 Vortex arrays 270
9.4 Experiments on vortices 273
9.5 Rapidly rotating condensates 275
9.6 Collective modes in a vortex lattice 280
Problems 286
References 288
10 Superfluidity 290
10.1 The Landau criterion 291
10.2 The two-component picture 294
10.2.1 Momentum carried by excitations 294
10.2.2 Normal fluid density 295
10.3 Dynamical processes 296
10.4 First and second sound 300
10.5 Interactions between excitations 307
10.5.1 Landau damping 308
Problems 314
References 315
11 Trapped clouds at non-zero temperature 316
11.1 Equilibrium properties 317
11.1.1 Energy scales 317
11.1.2 Transition temperature 319
11.1.3 Thermodynamic properties 321
11.2 Collective modes 325
11.2.1 Hydrodynamic modes above T 328
c
11.3 Collisional relaxation above T 334
c
11.3.1 Relaxation of temperature anisotropies 339
11.3.2 Damping of oscillations 342
Problems 345
References 346
12 Mixtures and spinor condensates 348
12.1 Mixtures 349
12.1.1 Equilibrium properties 350
12.1.2 Collective modes 354
12.2 Spinor condensates 356
Contents ix
12.2.1 Mean-field description 358
12.2.2 Beyond the mean-field approximation 360
Problems 363
References 364
13 Interference and correlations 365
13.1 Tunnelling between two wells 365
13.1.1 Quantum fluctuations 371
13.1.2 Squeezed states 373
13.2 Interference of two condensates 374
13.2.1 Phase-locked sources 375
13.2.2 Clouds with definite particle number 381
13.3 Density correlations in Bose gases 384
13.3.1 Collisional shifts of spectral lines 386
13.4 Coherent matter wave optics 390
13.5 Criteria for Bose–Einstein condensation 394
13.5.1 The density matrix 394
13.5.2 Fragmented condensates 397
Problems 399
References 399
14 Optical lattices 401
14.1 Generation of optical lattices 402
14.1.1 One-dimensional lattices 403
14.1.2 Higher-dimensional lattices 406
14.1.3 Energy scales 407
14.2 Energy bands 409
14.2.1 Band structure for a single particle 409
14.2.2 Band structure for interacting particles 411
14.2.3 Tight-binding model 416
14.3 Stability 418
14.3.1 Hydrodynamic analysis 421
14.4 Intrinsic non-linear effects 423
14.4.1 Loops 423
14.4.2 Spatial period doubling 427
14.5 From superfluid to insulator 431
14.5.1 Mean-field approximation 433
14.5.2 Effect of trapping potential 439
14.5.3 Experimental detection of coherence 439
Problems 441
References 442
x Contents
15 Lower dimensions 444
15.1 Non-interacting gases 445
15.2 Phase fluctuations 447
15.2.1 Vortices and the Berezinskii–Kosterlitz–Thouless transi-
tion 451
15.3 Microscopic theory of phase fluctuations 453
15.3.1 Uniform systems 455
15.3.2 Anisotropic traps 456
15.4 The one-dimensional Bose gas 460
15.4.1 The strong-coupling limit 461
15.4.2 Arbitrary coupling 466
15.4.3 Correlation functions 474
Problems 479
References 480
16 Fermions 481
16.1 Equilibrium properties 483
16.2 Effects of interactions 486
16.3 Superfluidity 489
16.3.1 Transition temperature 491
16.3.2 Induced interactions 496
16.3.3 The condensed phase 498
16.4 Pairing with unequal populations 506
16.5 Boson–fermion mixtures 508
16.5.1 Induced interactions in mixtures 509
Problems 511
References 513
17 From atoms to molecules 514
17.1 Bose–Einstein condensation of molecules 516
17.2 Diatomic molecules 518
17.2.1 Binding energy and the atom–atom scattering length 518
17.2.2 A simple two-channel model 520
17.2.3 Atom–atom scattering 526
17.3 Crossover: From BCS to BEC 527
17.3.1 Wide and narrow Feshbach resonances 528
17.3.2 The BCS wave function 530
17.3.3 Crossover at zero temperature 531
17.3.4 Condensate fraction and pair wave function 535
17.4 Crossover at non-zero temperature 540
17.4.1 Thermal molecules 540